Probability and Expected Value

QUESTION 1 Probability
(a) The expected value is a measure of central location(distribution) for a random
variable. It measures the population mean for a random variable.
The expected value for a discrete random variable is obtained by adding up the
product of the value of the random variable and its associated probability, taken over
all the values of the random variable (Newbold, Carlson & Thorne, 2013). That is,
.
For example, if we roll a normal six-sided die, the probability of landing on
1,2,3,4,5,or 6 is 1/6. To find the expected value when we roll a fair die we have:
(b) 1.
2. What were the average daily sales? 2.9 units
3. What was the probability of selling 2 or more loaves on any one day? 0.85
4. What was the probability of selling 3 or less? 0.65
5. What is the variance of the distribution? 6.67
6. What is the standard deviation? 2.58
Sales units
(x)
Number of
days (f) P(x) Exp Value
E(x) More than Less than [x-E(x)]^2 [x-E(x)]^2 P(x)
0 5 0.05 0 0.95 0.05 0 0
1 10 0.10 0.1 0.85 0.15 0.81 0.081
2 25 0.25 0.5 0.6 0.4 2.25 0.5625
3 25 0.25 0.75 0.35 0.65 5.0625 1.265625
4 20 0.20 0.8 0.15 0.85 10.24 2.048
5 15 0.15 0.75 0 1 18.0625 2.709375
Total 100 1.00 2.90 Variance 6.67
Average

daily sales
Total sales 290
Total days 100
Average 2.9
Standard
Deviation 2.58
Sales
units (x)
Number
of days
(f)
P(x)
Exp Value
E(x) More
than
Less
than [x-E(x)]^2 [x-E(x)]^2
P(x)
0 5 =C3/$C$9 =B3*D3 =1-D3 =D3 =(B3-E3)^2 =H3*D3
1 10 =C4/$C$9 =B4*D4 =F3-D4 =G3+D4 =(B4-E4)^2 =H4*D4
2 25 =C5/$C$9 =B5*D5 =F4-D5 =G4+D5 =(B5-E5)^2 =H5*D5
3 25 =C6/$C$9 =B6*D6 =F5-D6 =G5+D6 =(B6-E6)^2 =H6*D6
4 20 =C7/$C$9 =B7*D7 =F6-D7 =G6+D7 =(B7-E7)^2 =H7*D7
5 15 =C8/$C$9 =B8*D8 =F7-D8 =G7+D8 =(B8-E8)^2 =H8*D8
Total 100 =SUM(D3:D8) =SUM(E3:E8) Variance =SUM(I3:I8)
Average
daily
sales
Total
sales =B3*C3+B4*C4+B5*C5+B6*C6+B7*C7+B8*C8
Total
days =C9
Average =D12/D13
Standard
Deviation =SQRT(I9)
(c) What is the probability that a part selected at random:
1. Was produced by Machine W and should be reworked?

=
2. Was produced by Machine Z and is not satisfactory?
=
3. Was produced by Machine Y and should be scrapped?
=
4. Needs to be reworked?
=
5. Needs to be scrapped given that it was produced by machine W?
=
(d) μ = 4000; σ = 500.
1. What is the probability that sales will be greater than 4250 apples?
, where